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Universal Algorithms for Generalized Discrete Matrix Bellman Equations

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Universal Algorithms for Generalized Discrete Matrix Bellman Equations Universal algorithms for generalized discrete matrix Bellman equations with symmetric Toeplitz matrix∗ Serge˘ıSergeev† July 28, 2018 Abstract This paper presents two universal algorithms for generalized discrete matrix Bellman equations with symmetric Toeplitz matrix. The algorithms are semiring extensions of two well-known methods solving Toeplitz systems in the ordinary linear algebra. 1 Introduction As observed by B.A. Carr´e[1, 2], the Gaussian elimination without pivoting can be viewed as a prototype for some algorithms on graphs. M. Gondran [3] and G. Rote [4] made this observation precise by proving that the Gaussian elimination, under certain conditions, can be applied to the linear systems of equations over semirings. arXiv:math/0612309v1 [math.RA] 12 Dec 2006 The notion of universal algorithm over semiring was introduced by G.L. Litvinov, V.P. Maslov and E.V. Maslova in [5, 6]. These papers are to be considered in the framework of pub- lications [7, 8, 9, 10, 11, 12]) of the Russian idempotent school, and more generally, in ∗Research supported by RFBR grant 05-01-00824 and joint RFBR/CNRS grant 05-01-02807. †Department of Physics, Sub-Department of Quantum Statistics and Field Theory, M.V. Lomonosov Moscow State University, 119992 Leninskie Gory, GSP-2, Moscow, Russia. E-mail: [email protected] 1 the framework of idempotent and tropical mathematics, see [13, 14, 15] and references therein. Essentially, an algorithm is called universal if it does not depend on the com- puter representation of data and on a specific realization of algebraic operations involved in the algorithm [6]. Linear algebraic universal algorithms include generalized bordering method, LU- and LDM-decompositions for solving matrix equations. These methods are basically due to B.A. Carr´e, see also [6]. It was observed in [5, 6] that universal algorithms can be implemented by means of objective-oriented programming supported by C++, MATLAB, Scilab, Maple and other computer systems and languages. Such universal programs can be instrumental in many areas including the problems of linear algebra, optimization theory, and interval analysis over positive semirings, see [5, 11, 12, 16]. This paper presents new universal algorithms based on the methods of Durbin and Levin- son, see [17], Sect. 4.7. These algorithms solve systems of linear equations with symmet- ric Toeplitz matrices. Our universal algorithms have the same computational complexity O(n2) as their prototypes which beats the complexity O(n3) of the LDM-decomposition method. All algorithms are described as MATLAB-programs, meaning that they can be actually implemented. The author is grateful to G.L. Litvinov and A.N. Sobolevski˘ıfor drawing his attention to this problem and for valuable discussions. 2 Semirings and universal algorithms A set S equipped with addition ⊕ and multiplication ⊙ is a semiring (with unity) if the following axioms hold: 1) (S, ⊕) is a commutative semigroup with neutral element 0; 2) (S, ⊙) is a semigroup with neutral element 1 =6 0; 3) a ⊙ (b ⊕ c)=(a ⊙ b) ⊕ (a ⊙ c), (a ⊕ b) ⊙ c = (a ⊙ c) ⊕ (b ⊙ c) for all a, b, c ∈ S (distributivity); 2 4) 0 ⊙ a = a ⊙ 0 = 0 for all a ∈ S. In the sequel, we omit the notation ⊙ whenever this is convenient and does not lead to confusion. The semiring S is called idempotent if a ⊕ a = a for any a ∈ S. In this case ⊕ induces the canonical partial order relation a b ⇔ a ⊕ b = b. (1) The semiring S is called complete (cf. [18]), if any subset {xα} ⊂ S is summable and the infinite distributivity c ⊙ ( xα)= (c ⊙ xα), α α (2) x c x c . ( αLα) ⊙ = Lα( α ⊙ ) holds for all c ∈ S and {xα} ⊂ LS. This propertyL is natural in idempotent semirings and also in the theory of partially ordered spaces (cf. G. Birkhoff [19]) with partial order (1). Complete idempotent semirings are called a-complete (cf. [9]). Consider the closure operation ∞ a∗ = ai. (3) i=0 M In the complete semirings it is defined for all elements. The property a∗ = 1 ⊕ aa∗ = 1 ⊕ a∗a, (4) reveals that the closure operation is a natural extension of (1 − a)−1. We give some examples of semirings living on the set of reals R totally ordered by ≤: the semiring R+ with customary operations ⊕ = +, ⊙ = · and neutral elements 0 = 0 and 1 = 1; the semiring Rmax = R ∪ {−∞} with operations ⊕ = max ⊙ = +, and neutral elements 0 = −∞, 1 = 0; the semiring Rmax = Rmax ∪ {∞}, which is a completion of Rmax with the element ∞ satisfying a ⊕∞ = ∞ for all a, a ⊙∞ = ∞ ⊙ a = ∞ for a =6 0 b and 0 ⊙∞ = ∞ ⊙ 0 = 0; the semiring Rmax,min = R ∪ {∞} ∪ {−∞} with ⊕ = max, ⊙ = min, 0 = −∞, and 1 = ∞. 3 ∗ −1 Consider operation (3) for the examples above. In R+ the closure a equals (1 − a) if a< 1 and is undefined otherwise; in Rmax it equals 1 if a ≤ 1 and is undefined otherwise; ∗ ∗ ∗ in Rmax we have a = 1 for a ≤ 1 and a = ∞ for a> 1; in Rmax,min we have a = 1 for all a. Note that Rmax and Rmax are a-complete, so the closure is defined for any element b of these semirings. b The matrix operations ⊕ and ⊙ are defined analogously to their counterparts in linear algebra. Denote by Matmn(S) the set of all m × n matrices over the semiring S. By In we denote the n × n unity matrix, that is, the matrix with 1 on the diagonal and 0 off the 0 diagonal. As usual, we have AIn = InA = A and A = In for any A ∈ Matnn(S). The set Matnn(S) of all n × n square matrices is a semiring. Its unity is In and its zero is 0n, the square matrix with all entries equal to 0. If S is complete and/or idempotent, then so is ∗ the semiring Matnn(S). If S (and hence Matnn(S)) is complete, the closure A is defined for any matrix A and it satisfies (4). Note that if S is partially ordered, then Matmn(S) is ordered elementwise: A B iff Aij Bij for all i = 1,...,m and j = 1,...,n. If S is idempotent and canonically ordered (1), then the elementwise order of Matmn(S) also satisfies (1). The closure operation of matrices is important for the (discrete stationary) matrix Bellman equations X = AX ⊕ B. (5) If the closure of A exists and (4) holds, then X = A∗B is a solution to (5). In a-complete idempotent semirings the matrix A∗B is the least solution of this equation with respect to (1). Since A∗ is a generalization of (I − A)−1, the known universal algorithms for A∗ are generalizations of the methods for matrix inverses, and the known algorithms for Bellman equations are generalizations of the methods for AX = B. Further we consider the generalized bordering method. Let A be a square matrix. Closures of its main submatrices Ak can be found inductively. ∗ The base of induction is A1, the closure of the the first diagonal entry. Generally, we 4 represent Ak+1 as Ak gk A = , k+1 T hk ak+1 ! assuming that we have found the closure of Ak. In this representation, gk and hk are ∗ columns with k entries and ak+1 is a scalar. We also represent Ak+1 as U v A∗ k k . k+1 = T wk uk+1 ! Using (4) we obtain that T ∗ ∗ uk+1 =(hk Akgk ⊕ ak+1) , ∗ vk = A gkuk , k +1 (6) T T ∗ wk = uk+1hk Ak, ∗ T ∗ ∗ Uk = Akgkuk+1hk Ak ⊕ Ak. Consider the bordering method for finding the solution x = A∗b to (5), where X = x and (1) ∗ (k) B = b are column vectors. Firstly, we have x = A1b1. Let x be the vector found after (k − 1) steps, and let us write z x(k+1) = . xk+1 ! Using (6) we obtain that T (k) xk = uk (h x ⊕ bk ), +1 +1 k +1 (7) (k) ∗ (k+1) z = x ⊕ Akgkxk+1 . ∗ We have to compute Akgk. In general it makes a problem, but not in the case of the next section when A is symmetrical Toeplitz. We also note that the bordering method described by (6) and (7) is valid more generally over Conway semirings, see [18] for the definition. 5 3 Universal algorithms for Toeplitz linear systems Formally, a matrix A ∈ Matnn(S) is called (generalized) Toeplitz if there exist scalars r−n+1,...,r0,...,rn−1 such that Aij = rj−i for all i and j. Informally, Toeplitz matrices are such that their entries are constant along any line parallel to the main diagonal (and along the main diagonal itself). For example, r0 r1 r2 r4 r−1 r0 r1 r2 A = (8) r−2 r−1 r0 r1 r−3 r−2 r−1 r0 is Toeplitz. Such matrices are not necessarily symmetric. However, they are always persymmetric, that is, symmetric with respect to the inverse diagonal. This property is T algebraically expressed as A = EnA En, where En = [en,...,e1]. By ei we denote the 2 column whose ith entry is 1 and other entries are 0. The property En = In (where In is the n × n identity matrix) implies that the product of two persymmetric matrices is persymmetric. Hence any degree of a persymmetric matrix is persymmetric, and so is the closure of a persymmetric matrix. Thus, if A is persymmetric, then ∗ ∗ T EnA =(A ) En. (9) Further we deal only with symmetric Toeplitz matrices. Consider the equation y = (n) (n) T Tny ⊕ r , where r = (r1,...rn) and Tn is defined by the scalars r0,r1,...,rn−1 so that Tij = r|j−i| for all i and j. This is a generalization of the Yule-Walker problem [17].
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